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# Invariant Statistical Convergence of Sequences of Sets with respect to a Modulus Function

**Academic Editor:**Irena Rachůnková

#### Abstract

We introduce and study the concept of invariant convergence for sequences of sets with respect to modulus function and give some inclusion relations.

#### 1. Introduction

The concept of statistical convergence for sequences of real numbers was introduced by Fast [1] and studied by Šalát [2] and others. Let and . Then the natural density of is defined by , if the limit exists, where denotes the cardinality of .

A sequence complex numbers is said to be statistically convergent to if, for each ,

Convergence concept for sequences of set had been studied by Beer [3], Aubin and Frankowska [4], and Baronti and Papini [5]. The concept of statistical convergence of sequences of set was introduced by Nuray and Rhoades [6] in 2012. Ulusu and Nuray [7] introduced the concept of Wijsman lacunary statistical convergence of sequences of set. Similarly, the concepts of Wijsman invariant statistical and Wijsman lacunary invariant statistical convergence were introduced by Pancaroglu and Nuray [8] in 2013.

A function is called a modulus, if(1) if and if only if ;(2);(3) is increasing;(4) is continuous from the right at .

A modulus may be unbounded (e.g., , ) or bounded (e.g., ).

Modulus function was introduced by Nakano [9] in 1953. Ruckle [10] used in idea of modulus function to construct a class of FK spaces. Consider

The space is closely related to the space which is a space with , for all real .

Maddox [11] defined the following spaces by using a modulus function : where is space of all complex sequences.

Later, Connor [12] extended his definition by replacing the Cesaro matrix with an arbitrary nonnegative matrix summability method as follow:

#### 2. Definitions and Notations

Let be a mapping of the positive integers into itself. A continuous linear functional on , the space of real bounded sequences, is said to be an invariant mean or a mean, if and only if,(1), for all sequences with for all ;(2), where ;(3) for all .

The mappings are assumed to be one-to-one such that for all positive integers and , where denotes the th iterate of the mapping at . Thus, extends the limit functional on , the space of convergent sequences, in the sense that , for all . In case is translation mapping , the mean is often called a Banach limit and , the set of bounded sequences all of whose invariant means are equal, is the set of almost convergent sequences.

It can be shown that where,

Nuray and Savaş [13] defined the following sequence spaces by using a modulus function and a nonnegative regular matrix :

*Definition 1 (see [14]). *A set of positive integers is said to have uniform invariant density of zero if
uniformly in .

By using uniform invariant density, the following definition was given.

*Definition 2 (see [10]). *A complex number sequence is said to be -statistically convergent to if, for every ,

This will be denoted by or .

Let be a metric space. For any point and nonempty subset of , we define the distance from to by The concept of Wijsman convergence was introduced by Wijsman [7] as follows.

Let be a metric space. For any nonempty closed subsets , , we say that the sequence is Wijsman convergent to if for each . This will be denoted by .

Convergence concept for sequences of set had been studied by Beer [3], Aubin and Frankowska [4], and Baronti and Papini [5]. The concepts of Wijsman statistical convergence and Wijsman strong Cesaro summability were introduced by Nuray and Rhoades [6] as follows.

Let be a metric space. For any nonempty closed subsets , , the sequence is said to be Wijsman strongly Cesaro summable to if, for each ,

Let be a metric space. For any nonempty closed subsets , , the sequence is said to be Wijsman statistically convergent to if, for and each ,

In this case we write or .

#### 3. Main Result

The purpose of this paper is, by using a modulus function, to introduce and study new sequence spaces of sequences of sets. The following three definitions were given in [8].

*Definition 3. *Let be a metric space. For any nonempty closed subsets , , we say that the sequence is Wijsman invariant convergent to , if, for each ,
uniformly in .

In this case, we write and the set of all Wijsman invariant convergent sequences of sets will be denoted .

*Definition 4. *Let be a metric space. For any nonempty closed subsets , , we say that the sequence is Wijsman strongly invariant convergent to , if for each ,
uniformly in .

In this case, we write and the set of all Wijsman strongly invariant convergent sequences of sets will be denoted .

*Definition 5. *Let be a metric space. For any nonempty closed subsets , , we say that the sequence is Wijsman invariant statistically convergent to , if, for each and for each ,
uniformly in .

In this case, we write and the set of all Wijsman invariant statistically convergent sequences of sets will be denoted .

Let be metric space. For any nonempty closed subsets , and , we define the sequences of sets space as follows:

Now, by using a modulus function, we introduce the following new sequence spaces of sequences of sets.

*Definition 6. *Let be a metric space and be a modulus function. For any nonempty closed subsets , , we say that the sequence is Wijsman strongly invariant convergent to with respect to the modulus , if, for each ,
uniformly in .

In this case we write and the set of all Wijsman strongly invariant convergent sequences of sets with respect to the modulus will be denoted .

Let be metric space and be a modulus function. For any nonempty closed subsets , and , we define the sequences of sets space as follows:

If , then the spaces and reduce to and , respectively.

Now we study the relation between and convergence.

Theorem 7. *Let be a metric space. For any nonempty closed subsets , . Then*(i)* implies ;*(ii)* is bounded and implies ;*(iii)*, if is bounded.*

*Proof. * Let and . Then we can write
which yields the result.

Suppose that and is bounded, for each , set

Let and select such that
for all and , and set . Now, for all and , we have that

Hence, is strongly invariant convergent to with respect to the modulus function .

This is an immediate consequence of and .

This completes the proof of the theorem.

Theorem 8. *Let be a modulus function. Then .*

*Proof. *Suppose that, . Then we can write
where is an integer such that . Therefore, .

Theorem 9. *If is a modulus function and is strongly invariant convergent to , then is strongly invariant convergent to with respect to the modulus ; that is, .*

*Proof. *Let . Then we can write
uniformly in .

Let and choose with such that for . Write ,
where the first summation is over and second over . Then and, for ,
where denotes the integer part of . By definition modulus function, we have, for ,
Hence, , which together with yields .

Lemma 10 (see [15]). *Let be a modulus function. Let be given constant. Then, there is a constant such that .*

Theorem 11. *Let be a bounded sequence and be a modulus function. Then is Wijsman strongly invariant convergent to with respect to modulus if and only if is Wijsman strongly invariant convergent to ; that is,
**
where denotes the set of bounded sequences of sets.*

*Proof. *The proof of the theorem follows from Theorem 9 and Lemma 10.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### References

- H. Fast, “Sur la convergence statistique,”
*Colloquium Mathematicae*, vol. 2, no. 3-4, pp. 241–244, 1951. View at: Google Scholar | Zentralblatt MATH | MathSciNet - T. Šalát, “On statistically convergent sequences of real numbers,”
*Mathematica Slovaca*, vol. 30, no. 2, pp. 139–150, 1980. View at: Google Scholar | Zentralblatt MATH | MathSciNet - G. Beer, “Convergence of continuous linear functionals and their level sets,”
*Archiv der Mathematik*, vol. 52, no. 5, pp. 482–491, 1989. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. P. Aubin and H. Frankowska,
*Set Valued Analysis*, Birkhäuser, Boston, Mass, USA, 1990. View at: MathSciNet - M. Baronti and P. L. Papini, “Convergence of sequences of sets,” in
*Methods of Functional Analysis in Approximation Theory*, vol. 76, pp. 135–155, Birkhäuser, Basel, Switzerland, 1986. View at: Google Scholar | Zentralblatt MATH | MathSciNet - F. Nuray and B. E. Rhoades, “Statistical convergence of sequences of sets,”
*Fasciculi Mathematici*, no. 49, pp. 87–99, 2012. View at: Google Scholar | Zentralblatt MATH | MathSciNet - U. Ulusu and F. Nuray, “Lacunary statistical convergence sequences of sets,”
*Progress in Applied Mathematics*, vol. 4, no. 2, pp. 99–109, 2012. View at: Google Scholar - N. Pancaroglu and F. Nuray, “On invariant statistically convergence and lacu-nary invariant statistically convergence of sequences of sets,”
*Progress in Applied Mathematics*, vol. 5, no. 2, pp. 23–29, 2013. View at: Google Scholar - H. Nakano, “Concave modulars,”
*Journal of the Mathematical Society of Japan*, vol. 5, no. 1, pp. 29–49, 1953. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - W. H. Ruckle, “$FK$ spaces in which the sequence of coordinate vectors is bounded,”
*Canadian Journal of Mathematics*, vol. 25, pp. 973–978, 1973. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - I. J. Maddox, “Sequence spaces defined by a modulus,”
*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 100, no. 1, pp. 161–166, 1986. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Connor, “On strong matrix summability with respect to a modulus and statistical convergence,”
*Canadian Mathematical Bulletin*, vol. 32, no. 2, pp. 194–198, 1989. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. Nuray and E. Savaş, “Some new sequence spaces defined by a modulus function,”
*Indian Journal of Pure and Applied Mathematics*, vol. 24, no. 11, pp. 657–663, 1993. View at: Google Scholar | MathSciNet - E. Savaş, “Strongly $\sigma $-convergent sequences,”
*Bulletin of the Calcutta Mathematical Society*, vol. 81, no. 4, pp. 295–300, 1989. View at: Google Scholar | Zentralblatt MATH | MathSciNet - E. Savaş, “On strong almost $A$-summability with respect to a modulus and statistical convergence,”
*Indian Journal of Pure and Applied Mathematics*, vol. 23, no. 3, pp. 217–222, 1992. View at: Google Scholar | MathSciNet

#### Copyright

Copyright © 2014 Nimet Pancaroglu and Fatih Nuray. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.